Kelvin asked:

Is murder of innocent people wrong according to classical utilitarianism? If so explain why or why not.

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Any ethical theory that wants to be taken seriously will have the murder of innocents fall into the
"wrong" column. This includes utilitarianism. But since utilitarianism doesn't give absolute prohibitions
to specific kinds of acts, the technically correct answer is "almost always."

Classical utilitarians think that the right action is the action that maximizes the amount of happiness in
the world. The murder of innocents will not maximize happiness, for two fairly obvious reasons (there
may be more):

First, and most importantly, murdering an innocent person deprives that person of all of the future
happiness of his life. Second, if people were permitted to run around killing one another, then
everyone would live in constant fear of being killed. This, of course, is a pretty unhappy consequence.

Some people who have more deontological leanings (i.e., believe that things like "don't kill" should be
absolutemoral rules) aren't satisfied with the utilitarian account. They point out that under some
circumstances, it might actually maximize happiness to kill the innocent person — for example, if by
killing one person, we could save several innocent people's lives, or if by killing an innocent murder
suspect, we could increase the deterrent effect on future potential murders.

This is a serious challenge for utilitarians. There are responses available, but it takes some fairly
sophisticated arguments. The best source that's easily understood to follow up on the utilitarian
response that I know is Chapter Four of William Shaw's Contemporary Ethics: Taking Account of
Utilitarianism.

Jonathan Ichikawa